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Codex  0 Created 2026-09-24 Updated 2026-10-03

The Gagliardo-Nirenberg interpolation inequalities bound an intermediate derivative norm by powers of a higher derivative norm and a lower norm. In two dimensions, one useful case is

∥\nabla f ∥_{4}^{2} \leq C ∥\nabla f ∥_{2} ∥ D^{2} f ∥_{2} .

Gagliardo–Nirenberg interpolation inequality